On the use of Markov chains and PerronFrobenuis Theorem in Population Genetics

Ola HossjerDept. of MathematicsStockholm University

June 2015

Motivation and Outline

I Conservation biology: Protect genetic diversity in order toI Prevent inbreedingI Keep species viableI Improve environmental adjustment of species

I Outline:I Populations with substructureI Formulas for long term rate of loss of genetic variants

Effective population size Ne

I Population of size N

I Ne : Size of ideal population with same rate of loss of geneticvariants as studied population (smaller Ne → faster rate)

I Short term protection rule: Ne ≥ 50

I Ne/N varies between species

Genetic drift for size N populationI Discrete time t = 0, 1, 2, . . ..

I Genes g = 1, . . . , 2N.

I Two variants white and black

I νtg nr. of offspring of g , time t.

I Freq. of black, time t + 1, is

Xt+1 = 12N |{g ; g is black at t + 1}|

= 12N

∑g ;g black at time t νtg .

I {Xt} Markov chain, state space

X = {0, 12N , . . . ,

2N−12N , 1}

= {0} ∪ {1} ∪ { 12N , . . . ,

2N−12N }

= X0 ∪ X1 ∪ X2.

Absorbing states: X0, X1

Transient states: X2

2N = 10,Xt = 0.4,

Xt+1 = 0.5,νt2 = νt5 = νt6 = 0,νt1 = νt3 = νt4 = νt9 = νt,10 = 1,νt8 = 2,νt7 = 3.

Ideal population: Wright-Fisher (WF) model1Children pick parental genes independently;

{νtg}2Ng=1 ∼ Mult

(2N;

1

2N, . . . ,

1

2N

),

so that the transition kernel of {Xt} is

P = (P(x , y))x,y∈X ,

P(x , y) = P(Xt+1 = y |Xt = x) =(

2N2Ny

)x2Ny (1− x)2N(1−y).

Feller (1951) showed that the eigenvalues of P are

λ1 = λ2 = 1, λj =(2N − 1)(2N − 2) · · · (2N − j + 2)

(2N)j−2, j = 3, . . . , 2N+1,

so that the largest non-unit eigenvalue is

λ = λ3 = 1− 1

2N. (1)

It gives the asymtotic rate of fixation of one variant;

limt→∞

P(Xt ∈ X2)

λt= C , (0 < C <∞).

1Fisher (1921), Wright (1931).

Cannings modelCannings (1974) showed more generally that

λ1 = λ2 = 1, λj = E

(j−1∏g=1

νtg

), j = 3, . . . , 2N + 1,

if {νtg}2Ng=1 are exchangeable, so that in particular,

λ = λ3 = E (νt1νt2) = 1 + Cov(νt1, νt2) = 1− p,

with coalescence probability

p = −Cov(νt1, νt2)

= E [νt1(νt1−1)]2N−1

= 2N · E[(νt1

2

)]/(

2N2

)= P (two offspring have the same parent) .

Eigenvalue effective size2 NeE = size of a WF population with fixationrate λ:

λ = 1− 1

2NeE=⇒ NeE =

1

2(1− λ)=

N − 12

E [νt1(νt1 − 1)].

2Crow (1954), Ewens (1982).

Gene diversities

Introduce (predicted) gene diversities at time t = 0, 1, 2, . . .

Ht = 2Xt(1− Xt),= P(two genes picked with repl. have diff. variants|Xt)

ht = E (Ht)= P(two genes picked with repl. have diff. variants).

It can be shown thatht+1 = λht .

Hence gene diversities

ht = λth0 = (1− p)th0, (2)

tend to zero at the same multiplicative rate as non-fixationprobabilities P(Xt ∈ X2).

But gene diversities and coalescence probabilities (p) are easier toanalyze theoretically!!

Structured populationDivide into s subpopulations i = 1, . . . , s. Let

2Ni = number of genes in subpop. i , (∑

i Ni = N)Xti = fraction of black variants in subpop. i at time t,νtkig = nr. of offspring of gene g of subpop. k at time

t that end up in subpop. i at time t + 1,exchangeable for g = 1, . . . , 2Nk .

This gives dynamics

Xt+1,i =1

2Ni

s∑k=1

∑g ;g∈ subpop. k at time t,

g is black

νtkig .

Find, under suitable conditions,

NeE =1

2(1− λ),

where λ is rate of fixation, so that for some 0 < C <∞,

limt→∞

P(non-fixation at time t)

λt= C .

Structured population, contd.Let

Xt = (Xt1, . . . ,Xts).

If reproduction is time invariant, {Xt} is Markov chain with state space

X = {0, 12N1

, . . . , 2N1−12N1

, 1} × . . .× {0, 12Ns

, . . . , 2Ns−12Ns

, 1}= X0 ∪ X1 ∪ X2.

where

X0 = {(0, . . . , 0)},X1 = {(1, . . . , 1)},X2 = X \ (X0 ∪ X1),

and

P = (P(x, y))x,y∈X ,P(x, y) = P(Xt+1 = y|Xt = x),

λ = 3rd largest eigenvalue of P.

[c]

Gene diversities for structured populationIntroduce (predicted) gene diversities3 at time t = 0, 1, 2, . . .

Htij = Xti (1− Xtj) + (1− Xti )Xtj ,= P(two genes picked with repl. from i and j have diff. variants|Xt)

htij = E (Htij)= P(two genes picked with repl. from i and j have diff. variants)

between all pairs of subpopulations i and j . The column vector

ht = vec((htij)

si,j=1

)with s2 predicted gene diversities satisfies

ht+1 = Aht =⇒ ht = Ath0, (3)

whereA = (Aij,kl)ij,kl∈{1,...,s}×{1,...,s}

is a square matrix of order s2, and4

λ = λmax(A). (4)

3Maruyama (1970), Felsenstein (1972), Nei (1973).4By Perron-Frobenius Theorem applied to P.

Backward migration and coalescence theory to find A

If children pick parental subpopulations independently, with

Bik = probability by which genes in i pickparental subpopulations from k ∈ {1, . . . , s},

pijk = coalescence probability within k= P(two genes from i , j with parents in k , have same parent)

= Nk

2NiNjBikBjk

(E(νtki1(νtki1−1))

1− 12Ni

){i=j}

E (νtki1νtkj1){i 6=j}.

Then (3) holds, with

Aij,kl =

(1− 1

2Ni

){i=j}(

1− pijk

1− 12Nk

){k=l}

BikBjl . (5)

Motivation of (3) and (5)

We have that

ht+1,ij = P(genes from i and j at time t + 1 have different variants)= P(different genes picked from i and j at time t + 1)∑

k,l ·P(gene from i has parent from k at time t)

·P(gene from j has parent from l at time t)·P(different parents from k and l)·P(different variants of the different parents at time t)

= (1− 12Ni

){i=j}∑k,l BikBjl(1− pijk){k=l} · ht,kl/(1− 1

2Nk){k=l}

=∑

k,l Aij,klht,kl .

with Aij ,kl as in (5). In vector form this writes

ht+1 = Atht .

Computation of NeE

Subpop 1Subpop 2

Subpop 3

Subpop 4

Subpop 5

N4=400

N1=200

N3=50

N5=100

N2=400

5

2

10

3

4

252

5

(Ni )si=1 = (200, 400, 50, 400, 100),

N =∑s

i=1 Ni = 1150,

(Bik)si,k=1 =

0.94 0.05 0 0.01 00.0125 0.9825 0.005 0 00 0.1 0.82 0.08 00.005 0 0.0075 0.9875 00 0 0 0.05 0.95

,

Repr. = {(νtkig )2Nkg=1}

sk=1

∼ Mult(2Ni ;

Bi12N1, . . . , Bi1

2N1, . . . , Bis

2Ns, . . . , Bis

2Ns

)independently for i = 1, . . . , s,

pijk = 1/(2Nk),

NeE = 970,

Gene diversity effective size NeGLet

Wij = P(choose gene pair from i and j),

and collect them into row vector of length s2:

W = vec ((Wij)1≤i,j≤s)′ .

Predicted gene diversity for two randomly sampled genes at time t, is

ht = P(the two genes have different variants)=

∑1≤i,j≤s Wijhtij

= Wht

= WAth0.

It follows from (1) and (2), that for Wright-Fisher model

ht =

(1− 1

2N

)t

· h0. (6)

Gene diversity effective size over time interval [0, t] solves (6), i.e.

NeG ([0, t]) =1

2

[1−

(hth0

)1/t] t→∞→ 1

2(1− λ)= NeE .

Local/global NeG and NeE

Constant subpop. sizes

0 20 40 60 80 100

020

040

0600

800

100

0120

0

Time

Eff

ecti

veSiz

e

Subpop 1Subpop 2Subpop 3Subpop 4Subpop 5Subpop 1,2,3,4,5Asymptotic Limit

Solid - with replacementDashed - without replacement

Local bottleneck in 1

0 20 40 60 80 100

020

0400

600

800

100

0

Time

Eff

ecti

veSiz

eSubpop 1Subpop 2Subpop 3Subpop 4Subpop 5Subpop 1,2,3,4,5Asymptotic Limit

Solid - with replacementDashed - without replacement

Blocked migration 1-2

0 20 40 60 80 100

020

040

0600

800

100

0120

0

Time

Eff

ecti

veSiz

e

Subpop 1Subpop 2Subpop 3Subpop 4Subpop 5Subpop 1,2,3,4,5Asymptotic Limit

Solid - with replacementDashed - without replacement

Horizontal: NeE

Solid: t → NeG ([0, t]) for whole population and subpopulationsGlobal weights: Wij = 1/s2

Local weights, subpopulation k : Wij = 1{(i ,j)=(k,k)}

Proof of (4)

We have that

ht = Wht = WAth0 = Cλmax(A)t + o(λmax(A)t

)(7)

as t →∞. But also

ht =∑

ij Wijhtij=

∑ij WijE [Xti (1− Xtj) + Xtj(1− Xti )]

= E [φ(Xt)]= E [E (φ(Xt)|X0)]

=∑

x,y π(x)P(t)(x, y)φ(y),

(8)

where

φ(x) =∑

i ,j Wij [xi (1− xj) + xj(1− xi )] ,

π(x) = P(X0 = x),

Pt =(P(t)(x, y); x, y ∈ X

).

Perron-FrobeniusBlock decompose transition matrix as

P =

1 0 00 1 0

P20 P21 P22

=⇒ Pt =

1 0 00 1 0

P(t)20 P

(t)21 Pt

22

,

Since P22 is non-negative, irreducible and aperiodic, it has uniquelargest eigenvalue λ = λ3, and therefore

P(t)(x, y) = λtr(x)l(y) + o(λt), x, y ∈ X2, (9)

wherel = (l(x); x ∈ X2)r = (r(x); x ∈ X2)′

are left and right eigenvectors of P2 with eigenvalue λ andcomponents

l(x) > 0,r(x) > 0,

(10)

for all x ∈ X2.

Proof of (4), contd.

Use (8), (9), (10) and the fact that

φ(x) = 0, x ∈ X0 ∪ X1,φ(x) > 0, x ∈ X2 (if all Wij > 0),π(x) > 0, for some x ∈ X2,

to conclude

ht = λt∑

x,y∈X2π(x)r(x)l(y)φ(y) + o(λt)

= λt∑

x∈X2π(x)r(x) ·

∑y∈X2

l(y)φ(y) + o(λt)

= Cλt + o(λt),

with C > 0. Combining this with (7), we finally deduce

λ = λmax(A).

Other topicsI Various types of structure (geographic, age, sex, combinations, . . . ).

I Computer program GESP (Olsson et al, 2015).

I Large population asymptotics:

NeE =N

C1+ o(N) = NeC + o(N) as N →∞,

where NeC is coalescence effective size5 and C1 a coalescence rate.

I Small migration asymptotics:

NeE =N

C2B+ o(B−1) as B → 0,

where B = long term rate of subpopulation change in ancestral line.

I Leading right eigenvector of A to assess subpopulationdifferentiation6

5Nordborg and Krone (2002), Sjodin et al. (2005).6FST of Wright (1943), GST of Nei (1973).

References

Hossjer, O. (2011). Coalescence theory for a general class of structured populations with fast migration. Advancesin Probability Theory 43(4), 1027-1047.

Hossjer, O., Jorde, P.E. and Ryman, N. (2013). Quasi equilibrium approximations of the fixation index underneutrality: The island model. Theoretical Population Biology 84, 9-24.

Ryman, N., Allendorf, F.W., Jorde P.E., Laikre, L. and Hossjer, O. (2013). Samples from subdivided populationsyield biased estimates of effective size that overestimate the rate of loss of genetic variation. Molecular EcologyResources 14, 87-99.

Olsson, F. Hossjer, O., Laikre, L. and Ryman, N. (2013). Characteristics of the variance effective population sizeover time using an age structured model with variable size. Theoretical Population Biology 90, 91-103.

Hossjer, O. (2014). Spatial autocorrelation for subdivided populations with invariant migration schemes.Methodology and Computing in Applied Probability 16(4), 777-810. DOI 10.1007/s11009-013-9321-3.

Hossjer, O. and Ryman, N. (2014). Quasi equilibrium, variance effective population size and fixation index formodels with spatial structure. Journal of Mathematical Biology 69(5), 1057-1128. DOI 10.1007/s00285-013-0728-9.

Hossjer, O., Olsson, F., Laikre, L. and Ryman, N. (2014). A new general analytical approach for modeling patternsof genetic differentiation and effective size of subdivided populations over time. Mathematical Biosciences 258,113-133. DOI: 10.1016/j.mbs.2014.10.001.

Hossjer, O., Olsson, F., Laikre, L. and Ryman, N. (2015). Metapopulation Inbreeding Dynamics, Effective Size andSubpopulation Differentiation - a General Analytical Approach for Diploid Organisms. Theoretical PopulationBiology 102, 40-59. DOI:10.1016/j.tpb.2015.03.006.

Hossjer, O. (2015). On the eigenvalue effective size in structured populations. To appear in Journal ofMathematical Biology. Available online, DOI 10.1007/s00285-014-0832-5.

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